Pi Approximation Day

[Gerson W. Barbosa]

Since today is Pi Approximation Day, just another one for the collection. Not an outstanding approximation, but this might fit into the Exotic Ones category:

\(e^{\sqrt[11_{3}]{\frac{2222_{4}-\frac{1}{22_{9}^{2}+22_{16}^{2}+\frac{1}{1111_{7}}}}{99_{10}}}}\)

[Valentin Albillo]

.
Hi, all,

I can't let Pi Approximation Day (22/7) pass by without my humble contribution, so here you are.

Set FIX 6 and then run on your HP-42S (physical or virtual) this little piece of mystery code I've hastily concocted and see what you get:

01 LBL "?"
02 10
03 STO ST Y
04 LBL 00
05 RCL ST Y
06 RCL× ST Z
07 RCL+ ST Z
08 RCLx ST X
09 RCL× ST L
10 1/X
11 -
12 DSE ST Y
13 GTO 00
14 SQRT
15 END

See if you can fathom how/why it works.

Regards.
V.

[Thomas Klemm]

Nice.

(07-23-2022 01:20 AM)Valentin Albillo Wrote:  See if you can fathom how/why it works.

It uses Ramanujan's formula:

\(
\begin{align}
\pi^2 = 10 - \sum_{n=1}^{\infty} \frac{1}{n^3(n+1)^3}
\end{align}
\)

[J-F Garnier]

(07-23-2022 02:20 AM)Thomas Klemm Wrote:  

(07-23-2022 01:20 AM)Valentin Albillo Wrote:  See if you can fathom how/why it works.

It uses Ramanujan's formula:

Can't you let others have a chance to play?

J-F

[Albert Chan]

(07-22-2022 11:47 PM)Gerson W. Barbosa Wrote:  \(e^{\sqrt[11_{3}]{\frac{2222_{4}-\frac{1}{22_{9}^{2}+22_{16}^{2}+\frac{1}{1111_{7}}}}{99_{10}}}}\)

What is subscript / superscript notation mean?

[vaklaff]

(07-23-2022 10:56 AM)Albert Chan Wrote:  

(07-22-2022 11:47 PM)Gerson W. Barbosa Wrote:  \(e^{\sqrt[11_{3}]{\frac{2222_{4}-\frac{1}{22_{9}^{2}+22_{16}^{2}+\frac{1}{1111_{7}}}}{99_{10}}}}\)

What is subscript / superscript notation mean?

My wife says they are the radix (for example 16 as a subscript indicates a hexadecimal number) and the good old exponent.

[Gerson W. Barbosa]

(07-23-2022 11:26 AM)vaklaff Wrote:  

(07-23-2022 10:56 AM)Albert Chan Wrote:  What is subscript / superscript notation mean?

My wife says they are the radix (for example 16 as a subscript indicates a hexadecimal number) and the good old exponent.

Yes, they are base and exponent, respectively. In order to avoid confusion I should have written (22₉)² instead, for instance. That’s 20² = 400.

[J-F Garnier]

(07-23-2022 01:20 AM)Valentin Albillo Wrote:  Set FIX 6 and then run on your HP-42S (physical or virtual) this little piece of mystery code I've hastily concocted and see what you get:

01 LBL "?"
02 10
03 STO ST Y
04 LBL 00
05 RCL ST Y
06 RCL× ST Z
07 RCL+ ST Z
08 RCLx ST X
09 RCL× ST L
10 1/X
11 -
12 DSE ST Y
13 GTO 00
14 SQRT
15 END

Wonderful little program!
It uses the specific HP-42S stack capability, running it on my trusty 32S would require a few more steps. Of course the interest is in this 42S version.

Thank you Valentin, it's a pleasure to read you again.

J-F

[Gerson W. Barbosa]

(07-23-2022 07:00 AM)J-F Garnier Wrote:  

(07-23-2022 02:20 AM)Thomas Klemm Wrote:  It uses Ramanujan's formula:

Can't you let others have a chance to play?

I had found the general term 1/(x^2 + x)^3 when I saw Thomas Klemm’s post. After driving almost 400 km during the day, I was glad he was that fast :-)

Not as fun, but we can still rewrite Valentin’s intentionally slightly obfuscated program:

Code:


00 { 31-Byte Prgm }
01▸LBL "?"
02 0
03▸LBL 00
04 RCL ST Y
05 RCL× ST X
06 RCL+ ST L
07 X↑2
08 RCL× ST L
09 1/X
10 -
11 DSE ST Y
12 GTO 00
13 10
14 +
15 SQRT
16 END

1E1 -> 3,141592850081697528137175603520803
1E2 -> 3,141592653592821635772340191552596
1E3 -> 3,141592653589793270134930833666846
1E6 -> 3,141592653589793238462643383279535
pi —-> 3,141592653589793238462643383279503

Edited to fix a typo

[C.Ret]

(07-23-2022 11:44 AM)J-F Garnier Wrote:  

(07-23-2022 01:20 AM)Valentin Albillo Wrote:  Set FIX 6 and then run on your HP-42S (physical or virtual) this little piece of mystery code I've hastily concocted and see what you get:

01 LBL "?"
02 10
03 STO ST Y
04 LBL 00
05 RCL ST Y
06 RCL× ST Z
07 RCL+ ST Z
08 RCLx ST X
09 RCL× ST L
10 1/X
11 -
12 DSE ST Y
13 GTO 00
14 SQRT
15 END

Wonderful little program!

I'm trying to fit it to my HP-41C, but I must have missed something. Especially at step 09 RCL× ST L; where is the LAST register initialized on the first pass? Why isn't this a problem regardless of what is previously stored in the LAST register? Is this a specific HP-42S trick, does memory recall arithmetic modify the LAST register?

I'm stuck; need a little more beers to drink and any good advice to hear !

[Gerson W. Barbosa]

(07-23-2022 03:18 PM)C.Ret Wrote:  Is this a specific HP-42S trick, does memory recall arithmetic modify the LAST register?

Yes, it does, unlike memory store arithmetic:

05 RCL ST Y (T 10 10 10)
06 RCL× ST Z (… 100)
07 RCL+ ST Z (… 110)
08 RCL× ST X (… 12100)
09 RCL× ST L (… 1331000)

[C.Ret]

(07-23-2022 03:41 PM)Gerson W. Barbosa Wrote:  

(07-23-2022 03:18 PM)C.Ret Wrote:  Is this a specific HP-42S trick, does memory recall arithmetic modify the LAST register?

Yes, it does, unlike memory store arithmetic:

Thank you for your quick response; this is a different behavior from the HP-15C memory recall arithmetic logic. That explain why I was so lost.

001 LBL "WLP
002 10
003 ENTER^
004 LBL 00
005 RCL Y
006 X^2
007 LASTX
008 +
009 -3
010 Y^X
011 -
012 DSE Y
013 GTO 00
014 SQRT
015 END

With FIX 7 display format, this little program give a better approximation of PI than 22/07.
Surprisingly, the number of instructions is the same as Valentin Albillo's original code.

\( \sqrt{10-\frac{1}{\left ( 10^2+10 \right )^3}-\frac{1}{\left ( 9^2+9 \right )^3}-\frac{1}{\left ( 8^2+8 \right )^3}-\cdots -\frac{1}{\left ( 2^2+2 \right )^3}-\frac{1}{\left ( 1^2+1 \right )^3}} = \sqrt{ \frac{2\,780\,722\,699}{281\,746\,080} } \approx \pi \)

[Thomas Klemm]

(07-23-2022 11:59 AM)Gerson W. Barbosa Wrote:  Not as fun, but we can still rewrite Valentin’s intentionally slightly obfuscated program

Without recall-arithmetic it uses even fewer bytes:

Code:

00 { 27-Byte Prgm }
01▸LBL "?"
02 0
03▸LBL 00
04 RCL ST Y
05 ENTER
06 X↑2
07 +
08 ENTER
09 X↑2
10 ×
11 1/X
12 -
13 DSE ST Y
14 GTO 00
15 10
16 +
17 SQRT
18 END

Using a register we can translate it for other HP calculators as well.

HP-15C:

Code:

   001 {       44  0 } STO 0
   002 {       43 35 } g CLx
   003 {    42 21  0 } f LBL 0
   004 {       45  0 } RCL 0
   005 {          36 } ENTER
   006 {       43 11 } g x²
   007 {          40 } +
   008 {          36 } ENTER
   009 {       43 11 } g x²
   010 {          20 } ×
   011 {          15 } 1/x
   012 {          30 } −
   013 {    42  5  0 } f DSE 0
   014 {       22  0 } GTO 0
   015 {           1 } 1
   016 {           0 } 0
   017 {          40 } +
   018 {          11 } √x̅

HP-25:

Code:

01: 23 00    : STO 0
02: 00       : 0
03: 21       : x<->y
04: 31       : ENTER
05: 15 02    : g x^2
06: 51       : +
07: 31       : ENTER
08: 15 02    : g x^2
09: 61       : *
10: 15 22    : g 1/x
11: 41       : -
12: 24 00    : RCL 0
13: 01       : 1
14: 41       : -
15: 23 00    : STO 0
16: 15 61    : g x<>0
17: 13 04    : GTO 04
18: 22       : Rv
19: 01       : 1
20: 00       : 0
21: 51       : +
22: 14 02    : f SQRT
23: 13 00    : GTO 00

We can even do it on the HP-35 if we consider the following manual steps a program.

Initialize:

Code:

CLR
10  STO

Repeat until 0:

Code:

ENTER↑  ENTER↑  ×  +
ENTER↑  ENTER↑  ×  ×
1/x  -
RCL  1  -  STO

Finalize:

Code:

R↓
10  +
√x

---

While we could start subtracting the terms from an initial value of 10, it is better to start the summation from 0 to avoid cancellation.

[Didier Lachieze]

(07-23-2022 06:46 PM)Thomas Klemm Wrote:  We can even do it on the HP-35 if we consider the following manual steps a program.

For this, back in the days, you would have used an HP-35 Key Sequence Form.

[Gerson W. Barbosa]

(07-23-2022 06:46 PM)Thomas Klemm Wrote:  Without recall-arithmetic it uses even fewer bytes:

Code:

00 { 27-Byte Prgm }
01▸LBL "?"
02 0
03▸LBL 00
04 RCL ST Y
05 ENTER
06 X↑2
07 +
08 ENTER
09 X↑2
10 ×
11 1/X
12 -
13 DSE ST Y
14 GTO 00
15 10
16 +
17 SQRT
18 END

Thanks. I tend to value least number of steps over least number of bytes on the HP-42S where memory is not a concern, but I’ve just noticed there’s no reason at all to use recall arithmetic in my step 05 above when a simple X↑2 instruction suffices.

[Valentin Albillo]

.
Hi, Jean-François,

(07-23-2022 11:44 AM)J-F Garnier Wrote:  

(07-23-2022 01:20 AM)Valentin Albillo Wrote:  Set FIX 6 and then run on your HP-42S (physical or virtual) this little piece of mystery code I've hastily concocted and see what you get:

Wonderful little program!

Thank you very much for your appreciation.

Quote:It uses the specific HP-42S stack capability, running it on my trusty 32S would require a few more steps. Of course the interest is in this 42S version.

The little code I gave was intentionally obfuscated and particularized to add just 10 terms, to try and hide the "10" as if it were only the loop counter. My general non-obfuscated version which will add N terms is this 14-step, 26-byte HP-42S program:

01 LBL "?"
02 10
03 LBL 00
04 RCL ST Y
05 X↑2
06 RCL+ ST L
07 X↑2
08 RCL× ST L
09 1/X
10 -
11 DSE ST Y
12 GTO 00
13 SQRT
14 END

Key in N and XEQ "?". Running it on Free42 Decimal produces the following results for N=100, 1000, 10000 and 100000 terms:

      PI      3.14159 26535 89793 23846 26433 83279 502
      -------------------------------------------------
      100     3.14159 26535 92821 63577 23401 91552 596      (13 digits save 3 ulp)
      1000    3.14159 26535 89793 27013 49308 33666 846      (18 digits save 4 ulp)
      10000   3.14159 26535 89793 23846 29615 34056 207      (23 digits save 3 ulp)
      100000  3.14159 26535 89793 23846 26433 86462 443      (28 digits save 3 ulp)
      -------------------------------------------------
      PI      3.14159 26535 89793 23846 26433 83279 502

I like this series for Pi very much: it converges very fast (order 6), the general term is extremely simple to program needing just 6 steps, and this makes for speedy looping and very short running times, e.g. using as little as 100 terms gets you better than 12-digit accuracy and on Free42 it runs instantly.

Quote:Thank you Valentin, it's a pleasure to read you again.

Thank you, J-F, your feedback is most appreciated and if more people would do likewise that would surely motivate me to post again.

Best regards.
V.

[Thomas Klemm]

(07-23-2022 11:31 PM)Valentin Albillo Wrote:  Running it on Free42 Decimal produces the following results for N=100, 1000, 10000 and 100000 terms:

      PI      3.14159 26535 89793 23846 26433 83279 502
      -------------------------------------------------
      100     3.14159 26535 92821 63577 23401 91552 596      (13 digits save 3 ulp)
      1000    3.14159 26535 89793 27013 49308 33666 846      (18 digits save 4 ulp)
      10000   3.14159 26535 89793 23846 29615 34056 207      (23 digits save 3 ulp)
      100000  3.14159 26535 89793 23846 26433 86462 443      (28 digits save 3 ulp)
      -------------------------------------------------
      PI      3.14159 26535 89793 23846 26433 83279 502

This is the table I got using Valentin's program with the additional value for 1000000:

      PI      3.14159 26535 89793 23846 26433 83279 502
      -------------------------------------------------
      100     3.14159 26535 92821 63577 23401 91552 596      (13 digits save 3 ulp)
      1000    3.14159 26535 89793 27013 49308 33666 845      (18 digits save 4 ulp)
      10000   3.14159 26535 89793 23846 29615 34056 207      (23 digits save 3 ulp)
      100000  3.14159 26535 89793 23846 26433 86462 453      (28 digits save 3 ulp)
      1000000 3.14159 26535 89793 23846 26433 83283 745      (31 digits save 4 ulp)
      -------------------------------------------------
      PI      3.14159 26535 89793 23846 26433 83279 502

Please note the slight differences for 1000 and 100000.

This is the table I got using Gerson's program that adds 10 at the end:

      PI      3.14159 26535 89793 23846 26433 83279 502
      -------------------------------------------------
      100     3.14159 26535 92821 63577 23401 91552 596      (13 digits save 3 ulp)
      1000    3.14159 26535 89793 27013 49308 33666 846      (18 digits save 4 ulp)
      10000   3.14159 26535 89793 23846 29615 34056 207      (23 digits save 3 ulp)
      100000  3.14159 26535 89793 23846 26433 86462 443      (28 digits save 3 ulp)
      1000000 3.14159 26535 89793 23846 26433 83279 535      (33 digits save 3 ulp)
      -------------------------------------------------
      PI      3.14159 26535 89793 23846 26433 83279 502

Interestingly it agrees exactly with Valentin's table.

For 1000000 it shows the effect of cancellation when starting with 10.

---

Somewhat related: Programming Exercise (HP-15C, 15C LE - and others)

[Dan C]

I'm curious about the PI number...
How many decimals in the PI are needed to get acceptable results in practical everyday
calculations? Is 3.14 enough ? Or do you have to use more? ( i think you have, but how many?)

[Maximilian Hohmann]

Hello!

(07-24-2022 09:37 AM)Dan C Wrote:  How many decimals in the PI are needed to get acceptable results in practical everyday calculations? Is 3.14 enough ? Or do you have to use more? ( i think you have, but how many?)

For my own personal everyday applications involving Pi (like tinkering around the house and garden) 3.0 is a good enough approximation :-)

At work, many years ago, we were doing microvave imaging measurements and calculations. These involved propagation models with many e^i*Pi terms. Even using (then) state-of-the art measuring equipment (mostly from Hp of course, Agilent had not been sourced out then) no more then 4 (or maybe 5 for theoretical calculations) decimal places were required. Everything else would have been drowned in the noise anyway.

Regards
Max

[Dan C]

(07-24-2022 10:58 AM)Maximilian Hohmann Wrote:  Hello!

(07-24-2022 09:37 AM)Dan C Wrote:  How many decimals in the PI are needed to get acceptable results in practical everyday calculations? Is 3.14 enough ? Or do you have to use more? ( i think you have, but how many?)

For my own personal everyday applications involving Pi (like tinkering around the house and garden) 3.0 is a good enough approximation :-)

At work, many years ago, we were doing microvave imaging measurements and calculations. These involved propagation models with many e^i*Pi terms. Even using (then) state-of-the art measuring equipment (mostly from Hp of course, Agilent had not been sourced out then) no more then 4 (or maybe 5 for theoretical calculations) decimal places were required. Everything else would have been drowned in the noise anyway.

Regards
Max

Thanks for your input! So in real life calculations you are fine with PI at 3.1415, and that will do in most situations.